 Joukowski map from the upper half-disc to the upper half-plane

The holomorphic function $J(w)=-(w+w^{-1})$ maps $D_{+}=\{w:|w|<1,\mathrm{Im}w>0\}$ conformally and bijectively onto the upper half-plane. Its imaginary part is $\mathrm{Im}w(|w{|}^{-2}-1)>0$, and its derivative $w^{-2}-1$ never vanishes there.

Equality of two values implies $(w_1-w_2)(1-1/(w_1{w}_2))=0$; since $|w_1{w}_2|<1$, this proves injectivity. For any target $\zeta$ in the upper half-plane, the quadratic equation $w^2+\zeta w+1=0$ has two roots with product one and neither on the unit circle. Exactly one therefore has modulus below one; the imaginary-part formula forces that root to lie in $D_{+}$, proving surjectivity.

In particular, $z\mapsto e^{iz}$ maps $\{0<\mathrm{Re}z<\pi ,\mathrm{Im}z>0\}$ bijectively to $D_{+}$, and composition gives the conformal map $z\mapsto -2\cos z$ onto the upper half-plane. The two sides of the half-strip map to the outer real rays, while its bottom edge maps to the segment $[-2,2]$.